Expanding curves in $\mathrm{T}^1(\mathbb{H}^n)$ under geodesic flow and equidistribution in homogeneous spaces

Abstract

Let $H = \mathrm{SO}(n,1)$ and $A =\{a(t) : t \in \mathbb{R}\}$ be a maximal $\mathbb{R}$-split Cartan subgroup of $H$. Let $G$ be a Lie group containing $H$ and $\Gamma$ be a lattice of $G$. Let $x = g\Gamma \in G/\Gamma$ be a point of $G/\Gamma$ such that its $H$-orbit $Hx$ is dense in $G/\Gamma$. Let $\phi: I= [a,b] \rightarrow H$ be an analytic curve, then $\phi(I)x$ gives an analytic curve in $G/\Gamma$. In this article, we will prove the following result: if $\phi(I)$ satisfies some explicit geometric condition, then $a(t)\phi(I)x$ tends to be equidistributed in $G/\Gamma$ as $t \rightarrow \infty$. It answers the first question asked by Shah in ~\cite{Shah_1} and generalizes the main result of that paper.